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Multiplicative property of localized Chern characters for 2-periodic complexes

Jeongseok Oh

TL;DR

The paper proves a fundamental multiplicative property for localized Chern characters of $2$-periodic complexes, extending the framework to DM stacks and enabling a ring homomorphism from the $K$-group of periodic complexes to the bivariant Chow group. The core technique is a deformation-based proof that leverages a Grassmannian–graph construction and a specialization map, yielding a robust, deformation-friendly description of localized Chern characters. As a key application, the authors derive the functoriality of cosection-localized intersection homomorphisms by expressing these maps in terms of localized Chern characters and Todd classes, and by exploiting the multiplicativity to handle sums of vector bundles. These results advance the algebraic understanding of virtual fundamental classes and their functorial properties, with potential DM-stack generalizations and implications for localization in intersection theory.

Abstract

We prove the multiplicative property of localized Chern characters. As a direct consequence, a localized Chern character gives rise to a ring homomorphism from the K-group of periodic complexes to the bivariant Chow cohomology group. As an application, we prove the functoriality of Kiem-Li's cosection-localized intersection homomorphisms.

Multiplicative property of localized Chern characters for 2-periodic complexes

TL;DR

The paper proves a fundamental multiplicative property for localized Chern characters of -periodic complexes, extending the framework to DM stacks and enabling a ring homomorphism from the -group of periodic complexes to the bivariant Chow group. The core technique is a deformation-based proof that leverages a Grassmannian–graph construction and a specialization map, yielding a robust, deformation-friendly description of localized Chern characters. As a key application, the authors derive the functoriality of cosection-localized intersection homomorphisms by expressing these maps in terms of localized Chern characters and Todd classes, and by exploiting the multiplicativity to handle sums of vector bundles. These results advance the algebraic understanding of virtual fundamental classes and their functorial properties, with potential DM-stack generalizations and implications for localization in intersection theory.

Abstract

We prove the multiplicative property of localized Chern characters. As a direct consequence, a localized Chern character gives rise to a ring homomorphism from the K-group of periodic complexes to the bivariant Chow cohomology group. As an application, we prove the functoriality of Kiem-Li's cosection-localized intersection homomorphisms.

Paper Structure

This paper contains 8 sections, 8 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Multiplicative property of localized Chern characters
  3. Construction of localized Chern characters \ref{['Local:Ch']}
  4. Alternative description of \ref{['Alter:deC']}
  5. Proof of the multiplicative property (Theorem \ref{['Multi:Prop']})
  6. Functoriality of cosection-localized intersection homomorphisms
  7. Koszul 2-periodic complex
  8. Cosection-localized intersection homomorphism

Key Result

Theorem 1.1

Let $E^1_\bullet$, $E^2_\bullet$ be 2-periodic complexes on $Y$ strictly exact off $i_1:X_1 \hookrightarrow Y$, $i_2:X_2 \hookrightarrow Y$, respectively. Then the following multiplicative property holds,

Theorems & Definitions (16)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 6 more